Higher-order iterative methods free from second derivative for solving nonlinear equations

Abstract

Due to their importance, several numerical methods have been suggested and analyzed under certain conditions. These numerical methods have been constructed using different techniques such as Taylor series, homotopy perturbation method and its variant forms, quadrature formula, variational iteration method, and decomposition method (Chun, 2005, 2007; Ham et al., 2008; Javidi, 2009; Noor, 2006, 2010, 2010a; Noor et al. (2007, 2007a, 2007b). Using the technique of updating the solution and Taylor series expansion, Noor et al. (2007b) have suggested and analyzed a sixth-order predictor-corrector iterative type Halley method for solving the nonlinear equations. Ham et al. (2008) and Chun (2007) have also suggested a class of fifth-order and sixth-order iterative methods. In the implementation of the method of Noor et al. (2007b), one has to evaluate the second derivative of the function, which is a serious drawback of these methods. To overcome these drawbacks, we modify the predictor-corrector Halley